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Errors in Jacobian Elliptic Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg7629] Errors in Jacobian Elliptic Functions
• From: seanross at worldnet.att.net
• Date: Fri, 20 Jun 1997 16:16:02 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

I have an application which requires extensive use of Jacobian Elliptic
Functions like JacobiSN, EllipticPi and EllipticF.  I have found that
these functions have some very small errors near the ends of their
ranges.  For example, JacobiSN is supposed to be bounded by +1 and -1,
just like Sine and Cosine.  However, for JacobiSN[x,p], with p very
close to zero, JacobiSN can occasionally come out slightly greater than
+1.  The problem is that my application also requires that I take ArcSin
of JacobiSN so that numbers slightly greater than +1 return complex
values which mess everything up.  I have done some study on these errors
and they are not continuous functions of the arguments.  I can plot
regions as wide as 10^-5 where they have the error and then find
continuous regions with no errors.

Question #1.  Does anyone know if it is better to truncate the
JacobiSN's with something like Min[Max[JacobiSN[x,p],-1],1] or to
rescale the whole thing with something like JacobiSN[x,p]/(1+10^-11)?

The second manifestation of this error is in the EllipticPi and
EllipticF functions when the first argument is very large or the second
argument is very close to Pi/2.  I get obvious wrong answers for certain
arguments in this region and extraneous complex values in others.

Question #2.  Does anyone know of a way to estimate the error in an
ill-behaved function like this?  

Thanks, Sean Ross